The denominator of the last term in loop quantum gravity's equation [0.5*(4k+3)] will never be an integer, so it's not a valid unit fraction.
The {a,b,c} triplet I came up with for 4/(4k+3) is:
{ 2(k+1), 2(k+1), (k+1)(4k+3) }
This works too: { k+1, 2(k+1)(4k+3), 2(k+1)(4k+3) }...